A397000 - OEIS

A397000

Square array A(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. F^k(x), where F(x) is the e.g.f. of A396996.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 11, 31, 0, 1, 4, 21, 105, 370, 0, 1, 5, 34, 240, 1465, 6086, 0, 1, 6, 50, 454, 3880, 27322, 129264, 0, 1, 7, 69, 765, 8390, 82118, 644886, 3397296, 0, 1, 8, 91, 1191, 15950, 200059, 2167626, 18568790, 107221505, 0

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OFFSET

1,8

LINKS

Table of n, a(n) for n=1..55.

FORMULA

E.g.f. F(x) satisfies F'(x) = exp(F^l(x)), where F^l(x) denotes the l-th iterate of F, with F(0) = 0.

Let a(n,k,l) = n! * [x^n] F^k(x), where F^k(x) is the k-th iterate of F, and let b(n,k,l) = n! * [x^n] exp(F^k(x)).

a(n,0,l) = 0^(n-1), b(0,k,l) = 1,

a(n,k,l) = Sum_{j=0..n-1} binomial(n-1,j) * b(j,k+l-1,l) * a(n-j,k-1,l) for k > 0, and

b(n,k,l) = Sum_{j=1..n} binomial(n-1,j-1) * a(j,k,l) * b(n-j,k,l) for n > 0.

These recurrences are computationally expensive; in practice, direct iteration of the defining power series is much faster.